On the Theory of 1 - Motives

This is an overview and a preview of the theory of mixed motives of level ≤ 1 explaining some results, projects, ideas and indicating a bunch of problems. Dedicated to Jacob Murre Let k be an algebraically closed field of characteristic zero to start with and let S = Spec(k) denote our base scheme. Recall that Murre [46] associates to a smooth n-dimensional projective variety X over S a Chow cohomological Picard motive M(X) along with the Albanese motive M(X). The projector π1 ∈ CH(X × X)Q defining M(X) is obtained via the isogeny Pic(X) → Alb(X) between the Picard and Albanese variety, given by the restriction to a smooth curve C on X since Alb(C) = Pic(C) (such a curve is obtained by successive hyperplane sections). For a survey of classical Chow motives see [54] (cf. also [4]). In the case of curves M(X) is the Chow motive of X refined from lower and higher trivial components, i.e., M(X) and M(X), such that, for smooth projective curves X and Y (1) Hom(M(X),M(Y )) ∼= Hom(Pic(X),Pic(Y ))Q by Weil (see [58, Thm. 22 on p. 161] and also a remark of Grothendieck and Manin [41]). Furthermore, the semi-simple abelian category of abelian varieties up to isogeny is the pseudo-abelian envelope of the category of Jacobians and Q-linear maps. Thus, such a theory of pure motives of smooth projective curves is known to be equivalent to the theory of abelian varieties up to isogeny, as pointed out by Grothendieck: one-dimensional (pure) motives are abelian varieties. This formula (1) suggests that we may take objects represented by Picfunctors as models for larger categories of mixed motives of any kind of curves over arbitrary base schemes S. However, non representability of Pic for open schemes, forces to refine our models. Let X be a closure of X Date: December 12, 2005.